Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 642535, 5 pages
http://dx.doi.org/10.1155/2013/642535
Research Article
Strongly Almost Lacunary -Convergent Sequences
Adem KJlJçman
1
and Stuti Borgohain
2
1
Department of Mathematics and Institute for Mathematical Research, Faculty of Science, Universiti Putra Malaysia,
43400 Serdang, Selangor, Malaysia
2
Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India
Correspondence should be addressed to Adem Kılıc ¸man; kilicman@yahoo.com
Received 16 July 2013; Revised 12 September 2013; Accepted 2 October 2013
Academic Editor: S. A. Mohiuddine
Copyright © 2013 A. Kılıc ¸man and S. Borgohain. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We study some new strongly almost lacunary -convergent generalized diference sequence spaces defned by an Orlicz function.
We give also some inclusion relations related to these sequence spaces.
1. Introduction
Te notion of ideal convergence was frst introduced by
Kostyrko et al. [1] as a generalization of statistical convergence
which was later studied by many other authors.
By a lacunary sequence, we mean an increasing integer
sequence =(
) such that
0
=0 and ℎ
=
−
−1
→∞
as →∞.
Troughout this paper, the intervals determined by will
be denoted by
= (
−1
,
], and the ratio
/
−1
will be
defned by
.
An Orlicz function is a function :[0,∞)→[0,∞),
which is continuous, nondecreasing, and convex with (0)=
0, ()>0, for >0 and () → ∞, as →∞.
Let ℓ
∞
, , and
0
be the Banach space of bounded,
convergent, and null sequences =(
), respectively, with
the usual norm ‖‖= sup
|
|.
A sequence ∈ℓ
∞
is said to be almost convergent if all of
its Banach limits coincide. Let denote the space of all almost
convergent sequences.
Lorentz [2] introduced the following sequence space
={∈ℓ
∞
: lim
,
() exists uniformly in }, (1)
where
,
() = (
+
+1
+⋅⋅⋅+
+
)/(+1).
Te following space of strongly almost convergent
sequence was introduced by Maddox [3]:
[ ]={∈ℓ
∞
: lim
,
(|−|)
exists uniformly in for some },
(2)
where =(1,1,...).
Kızmaz [4] studied the diference sequence spaces ℓ
∞
(Δ),
(Δ), and
0
(Δ) of crisp sets. Te notion is defned as follows:
(Δ)={= (
):(Δ
)∈}, (3)
for =ℓ
∞
,, and
0
, where Δ=(Δ
) = (
−
+1
), for all
∈.
Te above spaces are Banach spaces, normed by
‖‖
Δ
=
1
+ sup
Δ
.
(4)
Tripathy et al. [5] introduced the generalized diference
sequence spaces which are defned as, for ≥1 and ≥1,
(Δ
)={=(
):(Δ
)∈}, for =ℓ
∞
,,
0
.
(5)
Tis generalized diference has the following binomial
representation:
Δ
=
∑
=0
(−1)
(
)
+
. (6)